Method of measuring the distance to a satellite in consideration of quantum and gravity effects, method of measuring a location using the same, and user terminal

ABSTRACT

A method of measuring a distance to a satellite, which is performed by an electronic device, according to an exemplary embodiment of the present invention, the method comprises receiving a linearly polarized photon from and angular momentum per unit mass of the satellite the satellite; measuring an amount of rotation of the polarized photon, the rotation being induced by a space-time warpage due to gravity; and calculating a distance to the satellite by using the rotation amount of the polarized photon and the angular momentum per unit mass of the satellite. The distance to the satellite may be calculated by the following equation, 
                 sin   ⁢     Θ   ⁡   (   r   )       ≅       -       l   obs         rr   s           ⁢       1   -       r   s     r             ,         
wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l obs ’ is the angular momentum per unit mass of the satellite, ‘r’ is the distance to the satellite, and ‘r s ’ is the Schwarzschild radius of the Earth.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority from and the benefit of Korean PatentApplications No. 10-2020-0096015, filed on Jul. 31, 2020, which ishereby incorporated by reference for all purposes as if fully set forthherein.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to a method of measuring the distance to asatellite, a method of measuring a location using the same, and a userterminal. More particularly, the present invention relates to a methodof measuring the distance to a satellite in consideration of quantum andgravity effects, a method of measuring a location using the same, and auser terminal.

Discussion of the Background

In the modern society, positioning technology such as GPS is widely usedin vehicle navigation and user terminals such as smartphones. Currently,in the technology used for GPS, position is measured by using thedistance from a user terminal to the satellite by receiving theelectromagnetic wave transmitted from the satellite by the user terminalon the ground. In more detail, measuring a position using atriangulation method, in which distances from three or more satellitesand coordinates of each satellite is used, is performed.

However, there is an error in distance measurement to a SATELLITE, sothe error of GPS up to now is relatively large.

The present invention provides a method of more accurately measuring adistance to a satellite through a rotation amount of polarized photonswhen passing through space-time warped by a gravity.

Even an experiment in the regime where a quantum system evolves onclassical curved space-time has never been fully assessed. Here, wedescribe a photon state with 1a unitary irreducible representation ofthe Wigner Rotation to investigate geometric phases induced bygravitational field between the ground station and the satellites in theEarth Orbits for various states of observers. It is found that there aregeneral relativistic, or classical, and the quantum component in theWigner rotation; When an observer is in a spiraling orbit, the quantumcomponent is obtained from 10-6 degree to 10-4 degree depending on thealtitude of the Earth Orbits, which should be measurable. This quantumrotation produced by the gravitational field would be the result ofintertwining of quantum and general relativity on the photon state andwould open up the road to test the gravitational effects on the quantumsystems.

Describing photon states observed by a moving observer (e.g., asatellite) in curved spacetime requires the understanding of bothquantum mechanics and general relativity, two essential branches ofmodern physics. One of the conceptual barriers for the relativistictreatment of quantum information is the difference of the role played bythe wave fields and the state vectors in relativistic quantum theory. Innon-relativistic quantum mechanics, the wave function of theSchrödinger's equation gives the probability amplitude which can be usedto define conserved particle densities or density matrices. However, itwas discovered that the relativistic equations are only indirectrepresentations for probability waves of a single particle. In 1939,Wigner proposed the idea that the quantum states of relativisticparticles can be formulated without the use of wave equations. He showedthat the states of a free particle are given by a unitary irreduciblerepresentation of the Poincare group. In Wigner's formulation,relativistic-particle states in different inertial frames are related bya little group element in the irreducible representation of Poincaregroup, called Wigner rotation.

While Wigner's original proposal was for the special relativity, therehave been several attempts to extend it to the domain of generalrelativity. Since extending Wigner's group to curved spacetime requiresthe standard directions (xyz) at every event, by introducing tetrads(frame fields) to define local coordinates, it has been shown thatmoving-particle states in curved spacetime are transformed into eachother by the Wigner rotation. For free-space QKD systems, it induces therotation of linear polarization of a photon observed between an earthground station and a satellite in the near-Earth orbit. Thus, it wouldbe a particularly important problem from not only a fundamental point ofview for testing general relativistic effects on quantum theory but alsoan application point of view for precision quantum metrology and freespace quantum communication.

In this work, it is demonstrated that the existence of non-trivialWigner rotation experienced by photons sent from the earth groundstation to a free-falling observer with non-zero angular momentum. Wemodel the spacetime around Earth with Schwarzschild spacetime wheretetrad fields can be globally defined as orientation-preservedcoordinate basis and we use the (− + + +) metric signature. Furthermore,it is also assumed34 that quantum field theories on spacetimes admit aspinor structure which will be employed for the quantum state of thephoton with given polarization. Considering that not much work has beendone on an experimental assessment of the regime in which quantumsystems evolve on classical curved space-time, our model could providethe test bed for probing the gravitational effects on the quantumsystem.

SUMMARY OF THE INVENTION

An object to be solved by the present invention is to provide a methodof measuring a distance to a satellite by using a rotation amount ofpolarized photon, which is caused by warp of space-time due to gravity.

Another problem to be solved by the present invention is to provide amethod of measuring a location by using the method of measuring adistance to a satellite.

Another problem to be solved by the present invention is to provide auser terminal for implementing the method of measuring a location.

The method of measuring a distance to a satellite, which is performed byan electronic device, according to an exemplary embodiment of thepresent invention, the method comprises receiving a linearly polarizedphoton from and angular momentum per unit mass of the satellite thesatellite; measuring an amount of rotation of the polarized photon, therotation being induced by a space-time warpage due to gravity; andcalculating a distance to the satellite by using the rotation amount ofthe polarized photon and the angular momentum per unit mass of thesatellite.

The distance to the satellite may be calculated by the followingequation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

The method of measuring a location according to an exemplary embodimentof the present invention comprises receiving, by an electronic device,from at least three or more satellites, a polarized photon of eachsatellite and angular momentum per unit mass of the satellites;measuring, by the electronic device, an amount of rotation of thepolarized photon of each satellite, the rotation being induced bywarpage of space-time due to gravity; calculating, by the electronicdevice, a distance to each satellite by using a rotation amount ofpolarization of each satellite and an angular momentum per unit mass ofeach satellite; and calculating a position relative to each of thesatellites by the electronic device by using the distance to each of thesatellites.

The distance to each satellite may be calculated by the followingequation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

For example, the electronic device may further receive a coordinated ofeach of the satellites from each of the satellites, when receiving thepolarized photon and angular momentum per unit mass of the satellites,and the electronic device further calculates a position of theelectronic device, when calculating a position relative to each of thesatellites.

The user terminal according to an exemplary embodiment of the presentinvention comprises a photon reception unit, a satellite informationreception unit, a polarization rotation measurement unit, and acalculation unit. The photon reception unit receives a polarized photonof each satellite from at least three or more satellites. The satelliteinformation reception unit receives an angular momentum per unit mass ofeach satellite from the at least three or more satellites. Thepolarization rotation measurement unit measures a rotation amount of thepolarized photon received by the photon receiving unit from eachsatellite. The calculation unit calculates a distance to each of thesatellites by using the rotation amount of the polarized photon, and theangular momentum per unit mass of each of the satellites, and calculatesrelative position of the user terminal by using the distance to each ofthe satellites.

The calculation units may calculate the distance to each of thesatellites by the following equation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

The satellite information reception unit may further receive acoordinate of each of the satellites from the each of the satellites,and the calculation unit may calculate a coordinates of the userterminal by using the coordinate of each of the satellites.

As described above, according to the present invention, it is possibleto more accurately measure the distance between the satellite and theuser terminal, thereby improving accuracy in a location measuring systemsuch as GPS.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawings will be provided by the Office upon request and paymentof the necessary fee.

FIG. 1A shows the Earth-Satellite system and corresponding coordinates.

FIG. 1B shows the wave vector and polarization of photon in each localframe.

FIG. 1C shows the wave vector and polarization of photon in the standardframe in comparison with FIG. 1B.

FIG. 2A and FIG. 2B show schematics of the trajectory of the observer(satellite) and photon with non-zero angular momentum in a spiralingorbit.

FIG. 3 shows the conceptual picture to interpret the tetrads for freefalling observer with non-zero angular momentum.

FIG. 4A and FIG. 4B show the classical part in orange solid line, thequantum part in blue solid line, and total WRA in green line for thecircular orbit.

FIG. 4C and FIG. 4D show the classical part in orange solid line, thequantum part in blue solid line, and total WRA in green line for thespiraling orbit.

FIG. 5 shows the relationship between the infinite classical rotationand quantum rotation versus affine parameters of a free-fall observerwith non-zero angular momentum for a circular orbit.

FIG. 6 is a flow chart showing a method for measuring a distance to asatellite according to an exemplary embodiment of the present invention.

FIG. 7 is a flow chart showing a method for measuring a positionaccording to an exemplary embodiment of the present invention.

FIG. 8 is a flow chart showing a method for measuring a locationaccording to another exemplary embodiment of the present invention.

FIG. 9 is a diagram showing a method of measuring relative positionsbetween satellites using distances from three satellites.

FIG. 10 is a block diagram showing a user terminal for implementing amethod for measuring a location according to the present invention.

DETAILED DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS

The present invention is described more fully hereinafter with referenceto the accompanying drawings, in which example embodiments of thepresent invention are shown. The present invention may, however, beembodied in many different forms and should not be construed as limitedto the example embodiments set forth herein. Rather, these exampleembodiments are provided so that this disclosure will be thorough andcomplete, and will fully convey the scope of the present invention tothose skilled in the art. In the drawings, the sizes and relative sizesof layers and regions may be exaggerated for clarity.

Hereinafter, exemplary embodiments of the present invention will bedescribed in detail.

Irreducible Representation of the Wigner Rotation

The Hilbert space vector of a photon is defined in a local inertialframe spanned by a tetrad, e_(ā) ^(μ)(x), â and μ=0, 1, 2, 3, whichsatisfies g_(μv)(x)=η_(â{circumflex over (b)})e^(â)_(μ)(x)e^({circumflex over (b)}) _(v) (X) and transforms in a way that

${{\overset{\_}{e}}_{\hat{a}}^{\mu}\left( \overset{\_}{x} \right)} = {\frac{\partial{\overset{\_}{x}}^{\mu}}{\partial x^{v}}{e_{\hat{a}}^{v}(x)}}$and ē_(â) ^(μ)(x)=Λ_({right arrow over (a)})^({right arrow over (b)})e_({right arrow over (b)}) ^(μ)(x) undergeneral coordinate and local Lorentz transformations, respectively. Avariation of a tetrad under an infinitesimal translation from x to x+δxis described by parallel transport to compare two vectors in a sametangent plane without a change of the vectors such thatδ(e _(â) ^(μ))=ē _(ā) ^(μ)(x+δx)−ē _(â) ^(μ)(x)→δx ^(λ)∇_(λ) e _(â)^(μ)(x).  (1.)

For the case that wave vector of a photon is measured in the observer'slaboratory, local covariant components of the wave vector,k_(â)(x)=e_(â) ^(μ)(x)k_(μ)(x), are changed along the photon's geodesicfrom x^(μ) to x^(μ)+k^(μ)(x)δξ such thatδk _(â)(x)=δ(e _(â) ^(μ)(x))k _(μ)(x)+e _(â) ^(μ)(x)δk _(μ)(x);δk_(μ)(x)=dξ∇ _(k) k _(μ)(x)   (2.)

Since a photon state in curved space-time follows a null geodesic in thegeometric optics limit^(13,14,31) and a local infinitesimal change of atetrad is antisymmetric^(25,29), Eq(2) can be rewritten ask _(â)(x)→k′ _(â)(x)≡k _(â)(x)+δk _(â)(x)=(δ_(â)^({circumflex over (b)})+λ_(â) ^({circumflex over (b)})(x)dξ)k_({circumflex over (b)})(x)=Λ_(â) ^({circumflex over (b)})(x)k_({circumflex over (b)})(x)   (3.),where λ_(â) ^({circumflex over (b)})(x)=(∇_(k)e_(â) ^(v) (x))e_(v)^({circumflex over (b)})(x) In other words, the effect of aninfinitesimal translation can be considered as an infinitesimal localLorentz transformation given by Λ_(â) ^({circumflex over (b)})(x)=δ_(â)^({circumflex over (b)})+λ_(â) ^({circumflex over (b)})(x). Throughoutthe paper, we use the hatted Latin and Greek letters for local inertialand general coordinates, respectively.

A Lorentz transformation, A, has the one-dimensional representations fora photon state with the helilcity, σ, given by³⁵

$\begin{matrix}\left. {{{{{{U(\Lambda)}\left. ❘{k,\sigma} \right\rangle} = {\sum\limits_{\sigma^{\prime}}^{}{D_{\sigma^{\prime}\sigma}\left( {W\left( {\Lambda,k} \right)} \right)}}}❘}\Lambda k},\sigma^{\prime}} \right\rangle & (4.)\end{matrix}$

W(Λ,k) is the Wigner's little group element, defined as W(Λ,k)=L⁻¹(Λk)ΛL(k) and D(W) is the irreducible representation of W. L(k) is theLorentz transformation such that L(p)k p. Accordingly, a displacement ofa photon state leads to a quantum phase called Wigner rotation angle(WRA). To get an explicit expression of the irreducible unitaryrepresentation of a Lorentz transformation, we use the canonical grouphomomorphism between the proper Lorentz group and its double cover,SL(2,

); a wave vector k of a photon is mapped to a Hermitian matrix K via Kσ_(â)k^(â), where σ_(â), â=1, 2, 3 are the Pauli matrices andσ_({circumflex over (0)}) is the 2×2 identity matrix. A Lorentztransformation is represented by the similarity transformation such thatAKA ^(†)=Λ^(μ) _(v) k ^(v)σ_(μ)  (5.)

with an element A of SL(2,

). The corresponding irreducible unitary representation of the littlegroup element for a massless particle is^(32,36)

$\begin{matrix}{{e^{i({\psi({\Lambda,k})})} = \left( \frac{{\left\lbrack {{\alpha\left( {1 + n^{3}} \right)} + {\beta n_{+}}} \right\rbrack b} + {\left\lbrack {{\gamma\left( {1 + n^{3}} \right)} + {\delta n_{+}}} \right\rbrack c^{*}}}{a\sqrt{b\left( {1 + n^{3}} \right)}} \right)^{2}},} & (6.)\end{matrix}$where Ψ(Λ,k) is the WRA. Detailed expressions for a, b, c, d, α, β, γ,and δ are given in the Supplementary Information (SI). Thus, A localinfinitesimal Lorentz transformation, Λ(x), leads to an infinitesimalWigner rotation (IWR) and the total Wigner rotation can be obtained by atime ordered integration of IWRs over the geodesic trajectory x(ξ) ofthe photon such that

$\begin{matrix}{{e^{i{\psi({\Lambda,_{n}^{r}})}} = {T{\exp\left\lbrack {i{\int{\psi\%\left( {{\Lambda\left( {x(\xi)} \right)},{n^{\hat{i}}(\xi)}} \right)d\xi}}} \right\rbrack}}},} & (7.)\end{matrix}$where n^(î)=k^(î)(x)/k^({circumflex over (0)})(x) and T is the timeordering operator. {tilde over (ψ)} and ψ are infinitesimal and totalWRA, respectively. In addition, it is well known that under a LT Λ, thepolarization vector, e_(ϕ) ^(â), is transformed in the standard framesuch thate _(ϕ′) ^(â) =R _({circumflex over (z)})(ψ(Λ,n ^(î)))e _(ϕ)^(â);ϕ′=ϕ+ψ(Λ,n ^(î))  (8.)

In the above formula, R_({circumflex over (z)})(ψ) represents therotation about {circumflex over (z)}-axis by the total WRA.

Model

In this work, we consider an Earth-satellite system depicted in FIGS. 1Ato 1C. The FIG. 1A shows the Earth-Satellite system and correspondingcoordinates; A photon is sent along its geodesic, which represented by ared line, and its polarization, represented by light-green arrows,measured in the local frame of a satellite. To compare the polarizationsmeasured at the surface of Earth and the satellite, we introduce thestandard frame in which a wave vector of the photon is aligned to thethird axis of observer's local frames (FIGS. 1B and 1C). We considerfollowing four cases: a stationary observer, a radially free-fallingobserver, a free-falling observer with non-zero angular momentum in acircular, and spiraling orbit (FIG. 2A). We note the Wigner rotation haszero angle in special relativity if the direction of boost and a wavevector both lie in the {circumflex over (x)}-{circumflex over (z)}plane, or the ŷ-{circumflex over (z)} plane. However, if a photon movesin the {circumflex over (x)}-{circumflex over (z)} plane and an observerin the {circumflex over (x)}-ŷ plane, WRA is not necessarily zero.Correspondingly, by the equivalence principle, all the observer isassumed to move in the plane ê_(r)-ê_(θ), i.e., the constant-y planewhile the photon's geodesic remains in the equatorial plane (FIG. 2B).FIGS. 2A and 2B show schematics of the trajectory of the observer(satellite) and photon. The geodesics of a photon traveling lies in theconstant φ-plane and the observers geodesics is lying in the equitorialplane, θ=π/2. In FIG. 2A, A, B, and C represents the geodesics ofmassive free-falling observers radially, in a circular orbit, andspiraling orbit, respectively. FIG. 2B shows the launching angle of aphoton.

We use Schwarzschild metric to model spacetime around Earth and choosespacelike components of the tetrads so that the first, second, and thirdaxis of the local frames become unit vectors of Schwarzschildcoordinates r, θ, and φ at infinity, i.e., ê_(â) ^(μ)(x_(∞))=ê_(b) ^(μ),where â=1, 2, and 3 correspond to b=r, θ, and φ, respectively. To definenon-spinning local frames, we apply Fermi-walker transport and paralleltransport conditions for the stationary and free falling observersrespectively. Detailed works are given in the supplementary material. Itis worth to mention that, when we set the local frame based onSchwarzschild coordinates, the rotation induced by the definition ofpolar coordinates has to be canceled out. In other words, since the unitvector of the coordinate r, ê_(r), is rotated as a coordinate φ changes,we choose φ-axis as the third axies of local frames to cancel out therotation when a wave vector is aligned to φ-axis for polarizationcomparison. Timelike components of the corresponding tetrads, ê_(t)^(μ), are set to the 4-velocity vector of a massive particle (e.g.satellite), moving along a geodesic corresponding to each case, todescribe a local frame of the observer. The 4-velocity vectors of theobservers and a wave vector of the photon are obtained in terms ofconserved quantities defined from killing vectors of Schwarzschildspacetime (SI #). We set conserved energy, ε_(photon), of a photon toits frequency to satisfy equivalence principle and the energy per unitmass, ε_(obs), of an observer to one in the unit where h=G=c=1 sinceε_(obs)=(1−r_(s)/r)dt/dτ; 1. Detailed works are given in theSupplementary Information (SI). We choose a launching angle of thephoton as 45° (FIG. 2B) and an angular momentum per unit mass of theobservers as

$0.4\sqrt{r_{s}r_{earth}}$so that the radial and polar components of the 4-velocity vectors havethe same value in Schwarzschild coordinate, ê_({circumflex over (0)})^(r) (x)=ê_({circumflex over (0)}) ^(θ)(x), where r_(earth) is theradius of Earth and r_(s) is the Schwarzschild radius.

Results

On observation of Eq. (8), it is noted that if every parameter is real,then the result of this equation is always real. In other words, theresult of Eq. (8) must be unity to avoid the imaginary Wigner angle.Accordingly, the first and second cases have zero WRAs since all theparameters are real. In the case that every parameter is not real,infinitesimal Winger rotation angle (IWA) is given by

$\begin{matrix}{{\overset{˜}{\psi} = {{2{Im}\left( \overset{˜}{\alpha} \right)} + {\frac{2n^{\overset{\hat{}}{1}}}{1 + n^{\overset{\hat{}}{3}}}{{Im}\left( \overset{\sim}{\beta} \right)}} + {\frac{2n^{\overset{\hat{}}{2}}}{1 + n^{\overset{\hat{}}{3}}}{{Im}\left( \overset{˜}{\gamma} \right)}}}},} & (9.)\end{matrix}$which corresponds to the last two cases, free falling observers withangular momentum. We note that IWA consists of a classical rotationaround the third axis, 2 Im({tilde over (α)}), and a quantum rotationinduced by the Wigner's little group elements,

${\frac{2n^{1}}{1 + n^{3}}{{Im}\left( \overset{\sim}{\beta} \right)}} + {\frac{2n^{2}}{1 + n^{3}}{{{Im}\left( \overset{\sim}{\gamma} \right)}.}}$For the circular-orbit case, parallel transport compensates the rotationinduced by spherical coordinates such that spacelike components of thetetrads are rotated by θ when observer moves by −rθ, leading to smallclassical WRA. For the spiraling-orbit case, parallel-transport rotatesthe tetrads around the local third-axis by 2Θ(r), which is defined as

$\begin{matrix}{{{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},} & (10.)\end{matrix}$where l_(obs) represents angular momentum per unit mass of an observer.In FIG. 3 , we show the conceptual picture to interpret the classicalIWA for the case of spiraling orbit. Since the leading term of classicalIWA depends on only the radial component of wave vector, see SI, thesystem can be simplified by neglecting the angular momentum of a photon;The tetrads are rotated around ê_(ϕ) by the gravity and becomesasymtotically identical to the Schwarzschild coordinate as the photon isobserved away from Earth. Accordingly, infinitesimal and total classicalWigner rotation is described by Θ(x+δx)−Θ(x) andΘ(x_(satellite))−Θ(x_(earth)), respectively.

TABLE 1 Observer in Circular Orbit Wigner angle (classical part +Classical Quantum Altitude quantum part) part part 300 km  2.42e−5−6.46e−14  2.42e−5 2000 km  9.64e−5 −3.03e−13  9.64e−5 20000 km −8.77e−7−7.018e−13  −8.77e−7 36000 km −9.93e−5 −7.61e−13 −9.93e−5 1.6 × 10¹¹ km(r = ∞) −6.25e−4 −8.02e−13 −6.25e−4 Observer in Spiraling Orbit Wignerangle Classical part − (classical part + Classical Rotation of QuantumAltitude quantum part) part tetrad part 300 km 1.13 1.16 −1.36e−9 5.32e−6 2000 km 6.30 6.31 −7.71e−9  2.41e−6 20000 km 24.48 24.48−2.25e−8 −1.80e−4 36000 km 29.31 29.31 −2.40e−8 −2.38e−4 1.6 × 10¹¹ km(r = ∞) 47.15 47.15 −9.15e−3 −3.54e−4

In Table 1, the rotation angles of the tetrads, classical part of IWA(general relativistic effect), 2 Im({tilde over (α)}), and the quantumrotation,

${{\frac{2n^{\overset{\hat{}}{1}}}{1 + n^{\overset{\hat{}}{3}}}{{Im}\left( \overset{\sim}{\beta} \right)}} + {\frac{2n^{\overset{\hat{}}{2}}}{1 + n^{\overset{\hat{}}{3}}}{{Im}\left( \overset{˜}{\gamma} \right)}}},$are compared for circular and spiraling orbits by integrating them fromthe surface of Earth to the altitudes of various Earth orbits. It isconfirmed that 2 Im({tilde over (α)}) represents the classical rotationby the almost identical two angles, 2Θ(r) and 2 Im({tilde over (α)}).Furthermore, it is shown that the circular orbit has much lesserclassical angle compared to spiraling case, as mentioned above. FIGS. 4Aand 4B show total WRA for the circular case. The classical part (generalrelativistic rotation) and the quantum part is represented by orangesolid line and blue solid line, respectively. The total WRA is shown inFIG. 4B. FIGS. 4C and 4D correspond to the spiraling case. Likewise, theclassical and quantum part is shown in FIGS. 4C and 4D show total WRA.For standard BB84 protocol, this corresponds to QBER (Quantum Bit ErrorRate) of 1.21% in the case of the LEO and 17.2% in the case of the MEO.This is consistent with a recent analysis, showing that anear-Earth-to-space QKD systems rely on entanglement distribution ofphoton states could have an additional contribution to its QBER as highas 0.7% because of spacetime curvature, and these effects are observablewith current technologies.

Conclusion and Discussion

In this work, we studied the Wigner rotation of a photon state inSchwarzschild spacetime to study a rotation of the polarization. Thegravitational field of Earth is described by the Schwarzchild metric.¹⁶We calculated the wave vector of the photon to get infinitesimal localLorentz transformations for the four cases of a stationary observer,free falling observer with zero angular momentum, and free fallingobserver with angular momentum in a circular and spiraling orbit. Forthe first two cases, the calculated Wigner angles are zeros. Wecalculate the non-zero Wigner angles for the last two cases in twodifferent ways: (1) by using approximations and (2) interpolationmethods for verification of our results since a differential equationfor the photon's trajectory is challenging to solve analytically and thetetrads of spiraling orbits have too complex forms to find physicalmeanings of them. It is found that two different approaches give thesame result up to 16 significant figures. The circular case results innon-zero WRA but its orders is only about 10⁻⁵ at NEO and LEO. For thespiraling case, quantum parts of WRA are 5.32×10⁻⁶° at NEO and−3.54×10⁻⁴° at infinity. These results are significantly larger thanprevious classical estimations. Furthermore, the total Wigner rotationshave angles of 1.13504° at NEO and 47.1469° at infinity and expected tocontribute QBER 1.21 and 17.2% to the quantum bit error rate in the caseof LEO and MEO, respectively. It is also interesting to compare theseresults with the works by Connors et al., who estimated the polarizationrotation angle of 82° at infinity from the X-rays near black hole inCygnus X-1 by using the general relativistic calculations. We believeour work would pave the road to test the gravitational effects on thequantum system.

Supplementary Material for Energy of Photon with the Affine ParameterUsed in this Specification (SI)

In general relativity, it is well known that the Lagrangian, L, can bechosen in the form (l)

$\begin{matrix}{\frac{1}{2}\left( \frac{ds}{d\xi} \right)^{2}} & ({S1})\end{matrix}$with the line element for the Schwarzschild metric, which is defined asfollows

$\begin{matrix}{{ds}^{2} = {{{- \left( {1 - \frac{r_{s}}{r}} \right)^{2}}{dt}^{2}} + {\left( {1 - \frac{r_{s}}{r}} \right)^{- 1}{dr}^{2}} + {r^{2}d\theta^{2}} + {r^{2}\sin^{2}\theta d{\phi^{2}.}}}} & ({S2})\end{matrix}$

If the Lagrangian has no dependence on specific coordinates (x^(μ)), theequations of motion imply the conservation of some quantities.Specifically, the equations of motion are written below as

$\begin{matrix}{{\frac{d}{d\tau}\left( \frac{\partial L}{\partial\left( {d{x^{\mu}/d}\xi} \right)} \right)} = {\frac{dL}{dx^{\mu}} = {0.}}} & ({S3})\end{matrix}$

From the above equation, the following identities hold, such as

$\begin{matrix}{\frac{\partial L}{\partial\left( {d{x^{\mu}/d}\xi} \right)} = {{g_{\mu\beta}\frac{dx^{\beta}}{d\xi}} = {{g_{a\beta}\delta_{\mu}^{\alpha}\frac{dx^{\beta}}{d\xi}} = {{g_{\mu\beta}\frac{\partial x^{\alpha}}{\partial x^{\mu}}\frac{{dx}^{\beta}}{d\xi}} = {{\frac{\partial}{\partial x^{\mu}} \cdot \frac{dx}{d\xi}} = {{const}.}}}}}} & ({S4})\end{matrix}$

In Schwarzschild spacetime, the time and azimuthal components of wavevector, k^(t), k^(ϕ), are constant, since time t, and azimuthal angle,ϕ, are cyclic coordinates in the metric. Therefore, from equation (S4),following two conserved quantities e and l are defined as (l)

$\begin{matrix}{{{e \equiv {{- \frac{\partial}{\partial t}} \cdot \frac{dx}{d\xi}}} = {\left( {1 - \frac{r_{s}}{r}} \right)\frac{dt}{d\xi}}},{{l \equiv {{- \frac{\partial}{\partial\phi}} \cdot \frac{dx}{d\xi}}} = {r^{2}\sin^{2}\theta{\frac{d\phi}{d\xi}.}}}} & ({S5})\end{matrix}$

Here, these two conserved quantities are called energy per unit restmass e at very large r, the distance from the origin and angularmomentum per unit rest mass at very low velocities, l, respectively. Forbrevity, we call ‘e’ the energy and ‘l’ the angular momentum in thispaper. For a photon, the geodesic equation in the Schwarzschild metriccan be rewritten as

$\begin{matrix}{{{{- \frac{e_{ph}^{2}}{\left( {1 - \frac{r_{s}}{r}} \right)}} + {\frac{1}{\left( {1 - \frac{r_{s}}{r}} \right)}\left( \frac{dr}{d\xi} \right)^{2}} + \frac{l_{ph}}{r^{2}}} = 0},} & ({S6})\end{matrix}$ $\begin{matrix}{{\left( \frac{dr}{d\xi} \right)^{2} = {{- \sqrt{e_{ph}^{2} - {\frac{l_{ph}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}} = {{- e_{ph}}\sqrt{1 - {\frac{b_{ph}^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}}}},} & ({S7})\end{matrix}$since all of the wave vectors of massless particles are null vectors. Inother words, we can get the explicit form of wave vectors andcorresponding dual vectors:

$\begin{matrix}{{k^{\mu}(x)} = \left( {\frac{e_{ph}}{1 - \frac{r_{s}}{r}},{{- e_{ph}}\sqrt{1 - {\frac{b_{ph}^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}},\frac{e_{ph}b_{ph}}{r^{2}},0} \right)} & ({S8})\end{matrix}$ $\begin{matrix}{{k_{\mu}(x)} = \left( {{- e_{ph}},{{- \frac{e_{ph}}{1 - \frac{r_{s}}{r}}}\sqrt{1 - {\frac{b_{ph}^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}},{e_{ph}b_{ph}},0} \right)} & ({S9})\end{matrix}$By the Equivalence principle, wave vectors in the local inertial frame,which is defined with radially free falling tetrads, should have thesame form with wave vectors in flat spacetime, which is as followsk _(â)(x)=(−ω,k _({circumflex over (1)}) ,k _({circumflex over (2)}) ,k_({circumflex over (3)})) where ω√{square root over (=(k_({circumflex over (1)}))²+(k _({circumflex over (2)}))²+(k_({circumflex over (3)}))²)}.   (S10)

In other words, inner product of wave vector with the time component oftetrads should be the same as the angular frequency of a photon in flatspacetime,

$\begin{matrix}{{k_{\hat{0}}(x)} = {{{{- \frac{e_{ph}}{1 - \frac{r_{s}}{r}}}\left( {1 - {\sqrt{\frac{r_{s}}{r}}\sqrt{1 - {\frac{b_{ph}^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}}} \right)} \cong {- {e_{ph}\left( {1 + \frac{r_{s}}{r} - {\sqrt{\frac{r_{s}}{r}}\sqrt{1 - {\frac{b_{ph}^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}}}} \right)}}} = {- {\omega.}}}} & ({S11})\end{matrix}$

Therefore, we conclude that photon's energy is the same as the frequencyof the photon, measured at the r=∞.

Parameter Dependences of the Wigner Rotation

A Hermitian matrix K, corresponding to each wave vector k of the photon,is defined as (2, 3)K=σ _(a) k ^(a)  (S12)where σ₀ is the 2×2 Identity matrix, and σ_(i) (i=1, 2, 3) are the Paulimatrices. Therefore, K has the form (2,3)

$\begin{matrix}{K = {{{k^{0}\begin{pmatrix}n^{3} & {n^{1} - {in^{2}}} \\{n^{1} + {in^{2}}} & n^{3}\end{pmatrix}}{where}n^{i}} = {\frac{k^{i}}{k^{0}}{\left( {{i = 1},2,3} \right).}}}} & ({S13})\end{matrix}$

Then, there is a matrix A in SL(2, C) for any Lorentz transformationsuch thatK′=Λ ^(a) _(b) k ^(b)σ_(a) =AKA ^(†)  (S14)

Since A is the elements of SL(2, C), it can be represented as

$\begin{matrix}{A = \begin{pmatrix}\alpha & \beta \\\gamma & \delta\end{pmatrix}} & ({S15})\end{matrix}$with unit determinant, i.e., αδ−γβ=1. For the Wigner's little groupelement (2-4) W(Λ,k)=L_(Λk) ⁻¹ΛL_(k), we can define a matrixcorresponding matrix S(Λ,k) in SL(2, C) such that W (Λ,k)=Λ(S(Λ,k)) andS(Λ,k)=A ⁻¹ _(k′) AA _(k)  (S16)where A_(k) corresponds to L(k) that {tilde over (k)} is transformedinto k. Here, {tilde over (k)}=(1,0,0,1) A_(k) has the form

$\begin{matrix}{{A_{k} = {\frac{1}{\sqrt{2{k^{0}\left( {1 + n^{3}} \right)}}}\begin{pmatrix}{k^{0}\left( {1 + n^{3}} \right)} & {- n_{-}} \\{k^{0}n_{+}} & {1 + n^{3}}\end{pmatrix}}}.} & ({S17})\end{matrix}$

If K′ is written by

$\begin{matrix}{K^{\prime} = {{{k^{\prime}}^{0}\begin{pmatrix}{1 + n^{\prime 3}} & n_{-}^{\prime} \\n_{+}^{\prime} & {1 - n^{\prime 3}}\end{pmatrix}} = {{AKA}^{\dagger} = {{{k^{0}\begin{pmatrix}b & c^{*} \\c & {a - b}\end{pmatrix}}{where}A} = \begin{pmatrix}\alpha & \beta \\\gamma & \delta\end{pmatrix}}}}} & ({S18})\end{matrix}$∈SL(2, C), then we get the following relations after some mathematicalmanipulations

$\begin{matrix}{a = {{\left( {{❘\alpha ❘}^{2} + {❘\gamma ❘}^{2}} \right)\left( {1 + n^{3}} \right)} + {\left( {{❘\beta ❘}^{2} + {❘\delta ❘}^{2}} \right)\left( {1 - n^{3}} \right)} + {\left( {{\alpha\beta^{*}} + {\gamma\delta}^{*}} \right)n_{-}} + {\left( {{\alpha^{*}\beta} + {\gamma^{*}\delta}} \right)n_{+}}}} & ({S19})\end{matrix}$ $\begin{matrix}{b = {{{❘\alpha ❘}^{2}\left( {1 + n^{3}} \right)} + {{❘\beta ❘}^{2}\left( {1 - n^{3}} \right)} + {\alpha\beta^{*}n_{-}} + {\alpha^{*}\beta n_{+}}}} & ({S20})\end{matrix}$ $\begin{matrix}{c = {{\alpha^{*}{\gamma\left( {1 + n^{3}} \right)}} + {\beta^{*}{\delta\left( {1 - n^{3}} \right)}} + {\beta^{*}\gamma n_{-}} + {\alpha^{*}\delta n_{+}}}} & ({S21})\end{matrix}$ $\begin{matrix}{{k^{\prime 0} = {\frac{a}{2}k^{0}}},\ {n^{\prime 3} = {\frac{2b}{a} - 1}},\ {n_{+}^{\prime} = {\frac{2c}{a}.}}} & ({S22})\end{matrix}$

Then S(Λ,k) has the form

$\begin{matrix}{S = {{\begin{pmatrix}e^{i{\psi/2}} & z \\0 & e^{{- i}{\psi/2}}\end{pmatrix}\psi} \in \left\lbrack {0,{4\pi}} \right\rbrack}} & ({S23})\end{matrix}$by direct calculation. Here, z is an arbitrary complex number.Substituting the equation (S20), in terms of a, b, α, β. γ, δ, into theequation (S18) we get the relation (2)

$\begin{matrix}{e^{i({{\psi({\Lambda,k})}/2})} = {\frac{{\left\lbrack {{\alpha\left( {1 + n^{3}} \right)} + {\beta n_{+}}} \right\rbrack b} + {\left\lbrack {{\gamma\left( {1 + n^{3}} \right)} + {\delta n_{+}}} \right\rbrack c^{*}}}{a\sqrt{b\left( {1 + n^{3}} \right)}}.}} & ({S24})\end{matrix}$

Moreover, the matrix S can be rewritten in the form

$\begin{matrix}{{S = \begin{pmatrix}e^{i{\psi/2}} & {e^{{- i}{\psi/2}}z} \\0 & e^{{- i}{\psi/2}}\end{pmatrix}},{\psi \in \left\lbrack {0,{4\pi}} \right\rbrack}} & ({S25})\end{matrix}$

The product of any two elements in this group becomes

$\begin{matrix}{{{S_{1}S_{2}} = \begin{pmatrix}e^{{- {i({\psi_{1} + \psi_{2}})}}/2} & {e^{{- {j({\psi_{1} + \psi_{2}})}}/2}\left( {z_{1} + {e^{i\psi_{1}}z_{2}}} \right)} \\0 & e^{{- {j({\psi_{1} + \psi_{2}})}}/2}\end{pmatrix}},{\psi \in {\left\lbrack {0,{4\pi}} \right\rbrack.}}} & ({S26})\end{matrix}$

In other words, we have the following composition law such that(z ₁,ψ₁)(z ₂,ψ₂)=(z ₁+exp(iψ ₁)z ₂,ψ₁+ψ₂).   (S27)

Thus, this group is the E(2) group.

There are two classes of the irreducible unitary representations of theE(2). One is the infinitesimal dimension representations and the otheris the one-dimension representations. However, the former has intrinsiccontinuous degrees of freedom. Therefore, the Lorentz transformation forthe photon has the one-dimension representations, since the photon isnot observed to have any continuous degrees of freedom. Therepresentations have the form (4)U(Λ)|k,σ

=e ^(iσψ(Λ,k))|Λk,σ

.  (S28)

Here, ψ(Λ,k) is the Wigner angle. When equation (S26) is expanded to thefirst order of dξ in the form

$\begin{matrix}{{{\left. e^{i{{\psi({\land {,k}})}/2}} \right.\sim 1} + {i{\overset{\sim}{\psi}\left( {\land {,k}} \right)}\frac{d\xi}{2}}},} & ({S29})\end{matrix}$the finite Wigner rotations can be built up as a time orderedintegration of infinitesimal Wigner rotations over the geodesictrajectory x(ξ) of the photon via

$\begin{matrix}{e^{i{{\psi({\Lambda,\overset{\rightarrow}{n}})}/2}} = {T{\exp\left\lbrack {i{\int{{\overset{\sim}{\psi}\left( {{\Lambda(\xi)},{\overset{\rightarrow}{n}(\xi)}} \right)}\frac{d\xi}{2}}}} \right\rbrack}}} & ({S30})\end{matrix}$where {right arrow over (n)}(ξ)={right arrow over (n)}(x(ξ)),Θ^(μ) _(v)(ξ)=ι^(μ) _(v) (x(ξ)) and T is the time order operator.

If the homogeneous Lorentz transformation is expressed as Λ^(a)_(b)=δ^(a) _(b)+ω^(a) _(b), the Wigner angle is related to ω^(μ) _(v).To see this, we expand A in terms of dξ as

$\begin{matrix}{A = {\begin{pmatrix}\alpha & \beta \\\gamma & \delta\end{pmatrix} = {{I + {\overset{\sim}{A}d\xi}} = {I + {\begin{pmatrix}\overset{\sim}{\alpha} & \overset{\sim}{\beta} \\\overset{\sim}{\gamma} & \overset{\sim}{\delta}\end{pmatrix}d{\xi.}}}}}} & ({S31})\end{matrix}$

By the condition that the A has unit determinant, {tilde over(δ)}=−{tilde over (α)}. In other words, the A is expanded in the form

$\begin{matrix}{A = {\begin{pmatrix}\alpha & \beta \\\gamma & \delta\end{pmatrix} = {{I + {\overset{\sim}{A}d\xi}} = {I + {\begin{pmatrix}\overset{\sim}{\alpha} & \overset{\sim}{\beta} \\\overset{\sim}{\gamma} & {- \overset{\sim}{\alpha}}\end{pmatrix}d{\xi.}}}}}} & ({S32})\end{matrix}$

Substituting the equation (S32) into the equation (S19), multiplyingσ_(a) both sides, and then taking a trace on both sides, we can get thefollowing equations by the relationtr(σ_(a)σ_(b))=2δ_(ab)ω^(a) _(b)=½δ^(ac) tr(σ_(b)σ_(c) Ã+σ _(c)σ_(b) Ã ^(†))  (S33)where tr(A) is the trace of A. From the equation (S33), we obtain{tilde over (α)}=½(ω⁰ ₃ +iω ¹ ₂){tilde over (β)}=½[(ω⁰ ₁−ω³ ₁)+i(ω⁰ ₂+ω² ₃)]{tilde over (γ)}=½[(ω⁰ ₁−ω³ ₁)−i(ω⁰ ₂−ω² ₃)].  (S34)

Real Parameters

The α, β, γ, δ, |β|², |β|², |γ|², |δ|² have the following forms by thedefinitionα=1+{tilde over (α)}dξ,β={tilde over (β)}dξ,γ={tilde over (γ)}dξ,δ=1−{tilde over (α)}dξ,|α|²=1+2{tilde over (α)}dξ,|β|²=0,|γ|²=0,|δ|²=1−2{tilde over (α)}dξ.  (S35)

Substituting n_(±)=n¹ into equations (S19), (S20), and (S21), we obtaina=2+2(2{tilde over (α)}n ³+({tilde over (β)}+{tilde over (γ)})n¹)dξ,  (S36)b=(1+n ³)+2(2{tilde over (α)}(1+n ³){tilde over (β)}n ¹)dξ.  (S37)and c=n ¹+({tilde over (γ)}(1+n ³)+{tilde over (β)}(1−n ³))dξ.  (S38)

In this work, we have calculated to the first order of dξ. Using theseparameters, the numerator of equation (S25) has the form[α(1+n ³)+βn ₊]b+[γ(1+n ³)+dn ₊]c*=2(1+n ³)+[2{tilde over (α)}+6{tildeover (α)}n ³+4α(n ³)²+4βn ¹+2βn ¹ n ³+2{tilde over (γ)}n ¹+2{tilde over(γ)}n ¹ n ³]dξ.   (S39)

We also have

$\begin{matrix}{\frac{1}{a\sqrt{b\left( {1 + n^{3}} \right)}} = {\frac{1}{2\left( {1 + n^{3}} \right)} - {\frac{{2\overset{\sim}{\alpha}n^{3}} + {\overset{\sim}{\beta}n^{1}} + {\overset{\sim}{\gamma}n^{1}} + {2\overset{\sim}{\alpha}} + {2\overset{\sim}{\beta}{n^{1}\left( {1 + n^{3}} \right)}^{- 1}}}{2\left( {1 + n^{3}} \right)}d{\xi.}}}} & ({S40})\end{matrix}$

By direct calculations, one can show that equation (S29) becomese ^(i(ψ(Λ,k)/2))=1.  (S41)

Therefore, it is evident that the observer who is at rest and fallingfree with zero angular momentum sees no Wigner rotation.

Complex Parameters

In the case that every parameter is not real, |α|², |β|², |γ|², |δ|²have the form|α|²=1+2Re({tilde over (α)})dξ|β|²=0|γ|²=0γδ|²=1−2Re({tilde over (α)})dξ  (S42)

where Re({tilde over (α)}) is the real part of the complex number {tildeover (α)}. Equations (S19), (S20), and (S21) are then rewritten

$\begin{matrix}{a = {{{\left( {{❘\alpha ❘}^{2} + {❘\gamma ❘}^{2}} \right)\left( {1 + n^{3}} \right)} + {\left( {{❘\beta ❘}^{2} + {❘\delta ❘}^{2}} \right)\left( {1 - n^{3}} \right)} + {\left( {{\alpha\beta}^{*} + {\gamma\delta}^{*}} \right)n_{-}} + {\left( {{\alpha^{*}\beta} + {\gamma^{*}\delta}} \right)n_{+}}} = {2 + {\left\lbrack {{4n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}} + {2{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}}} \right\rbrack d\xi}}}} & ({S43})\end{matrix}$ $\begin{matrix}{\begin{matrix}{b = {{{❘\alpha ❘}^{2}\left( {1 + n^{3}} \right)} + {{❘\beta ❘}^{2}\left( {1 - n^{3}} \right)} + {{\alpha\beta}^{*}n_{-}} + {\alpha^{*}\beta n_{+}}}} \\{= {{\left( {1 + {2{{Re}\left( \overset{\sim}{\alpha} \right)}d\xi}} \right)\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}d\xi}}} \\{{= {\left( {1 + n^{3}} \right) + {\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}}} \right\rbrack d\xi}}},}\end{matrix}} & ({S44})\end{matrix}$ $\begin{matrix}{and} & \end{matrix}$ $\begin{matrix}{\begin{matrix}{c = {{\alpha^{*}{\gamma\left( {1 + n^{3}} \right)}\beta^{*}{\delta\left( {1 - n^{3}} \right)}} + {\beta^{*}\gamma n_{-}} + {\alpha^{*}\delta n_{+}}}} \\{= {{{\overset{\sim}{\gamma}\left( {1 + n^{3}} \right)}d\xi} + {{\overset{\sim}{\beta^{*}}\left( {1 - n^{3}} \right)}d\xi} + {\left( {1 + {{\overset{\sim}{\alpha}}^{*}d\xi}} \right)\left( {1 - {\overset{\sim}{\alpha}d\xi}} \right)n_{+}}}} \\{= {n_{+} + {\left\lbrack {{\overset{\sim}{\gamma}\left( {1 + n^{3}} \right)} + {{\overset{\sim}{\beta}}^{*}\left( {1 - n^{3}} \right)} - {2in_{+}{{Im}\left( \overset{\sim}{\alpha} \right)}}} \right\rbrack d{\xi.}}}}\end{matrix}} & ({S45})\end{matrix}$

where Im({tilde over (α)}) is the imaginary part of the complex number{tilde over (α)}.

From above equations, we obtain

$\begin{matrix}{\left. {{{\left\lbrack {{\alpha\left( {1 + n^{3}} \right)} + {\beta n_{+}}} \right\rbrack b} + {\left\lbrack {{\gamma\left( {1 + n^{3}} \right)} + {\delta n}_{+}} \right\rbrack c^{*}}} = {\left( {\left( {1 + n^{3}} \right) + {\left\lbrack {{\overset{\sim}{\alpha}\left( {1 + n^{3}} \right)} + {\overset{\sim}{\beta}n_{+}}} \right\rbrack d}} \right.\text{⁠}\xi}} \right)\text{⁠}{\left( {\left( {1 + n^{3}} \right) + {{{\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}d\xi}} \right) + {\left( {n_{+} + {\left\lbrack {{\overset{\sim}{\gamma}\left( {1 + n^{3}} \right)} - {\overset{\sim}{\alpha}n_{+}}} \right\rbrack d\xi}} \right)\left( {n_{-} + {\left\lbrack {{{\overset{\sim}{\gamma}}^{*}\left( {1 + n^{3}} \right)} + {\overset{\sim}{\beta}\left( {1 - n^{3}} \right)} + {2{in}_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}}} \right\rbrack d\xi}} \right)}} = {\left( {1 + n^{3}} \right)^{2} + {n_{+}n_{-}} + \left\lbrack \text{⁠}{{\overset{\sim}{\alpha}\left( {1 + n^{3}} \right)}^{2} + {\overset{\sim}{\beta}{n_{+}\left( {1 + n^{3}} \right)}} + {2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)} - {\overset{\sim}{\alpha}n_{+}n_{-}} + {2{{Re}\left( \text{⁠}{{\overset{\sim}{y}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {\overset{\sim}{\beta}{n_{+}\left( {1 - n^{3}} \right)}} + {\left. {2{in}_{+}n_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}} \right\rbrack\text{⁠}{{{d\xi} = {{2\left( {1 + n^{3}} \right)} + \left\lbrack {{{2\overset{\sim}{\alpha}{n^{3}\left( {1 + n^{3}} \right)}} + {2\overset{\sim}{\beta}n_{+}} + {2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {{{Re}\left( {{\overset{\sim}{\gamma}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {2{in}_{+}n_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}d\xi}} = {2\left( {1 + n^{3}} \right)\left( {1 + \left\lbrack {\frac{{\left( {{2\overset{\sim}{\alpha}{n^{3}\left( {1 + n^{3}} \right)}} + {2\overset{\sim}{\beta}n_{+}} + {2{{Re}\left( \overset{\sim}{\alpha} \right.}}} \right)\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( \overset{\sim}{{\beta n}_{+}} \right)}\left( {1 + n^{3}} \right)} + {{{Re}\left( {{\overset{\sim}{\gamma}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {2{in}_{+}n_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}}}{\left( {2\left( {1 + n^{3}} \right)} \right.}d\xi} \right\rbrack} \right)}} \right.}}}}} \right.}}}} \right.}} & \left( {{S4}6} \right)\end{matrix}$ $\begin{matrix}{and} & \end{matrix}$ $\begin{matrix}{\frac{1}{a\sqrt{b\left( {1 + n^{3}} \right)}} = {\frac{1}{\left( {2 + {\left\lbrack {{4n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}\overset{\sim}{\gamma}n_{-}} \right)}}} \right\rbrack d\xi}} \right)\sqrt{\left( {1 + n^{3}} \right)^{2} + {\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)}} \right\rbrack d\xi}}} = {{\frac{1}{2}\left( {1 - {\frac{\left\lbrack {{4n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}} + {2{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}}} \right\rbrack}{2}d\xi}} \right)\frac{1}{\left( {1 + n^{3}} \right)}\frac{1}{\sqrt{\frac{1 + {\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}}} \right\rbrack d\xi}}{\left( {1 + n^{3}} \right)}}}} = {\frac{1}{2\left( {1 + n^{3}} \right)}\left( {1 - {\frac{\left\lbrack {{4n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}\left( {1 + n^{3}} \right)}} \right\rbrack}{2\left( {1 + n^{3}} \right)}d\xi}} \right)\left( {1 - {\frac{\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}}} \right\rbrack}{2\left( {1 + n^{3}} \right)}d\xi}} \right)}}}} & ({S47})\end{matrix}$

In the previous section, we proved real components of the parameterslead Eq. (S41) to one.

Substituting these results into (S41), we have the form

$\begin{matrix}{e^{i({{\psi({\land {,k}})}/2})} = {\frac{{\left\lbrack {{\alpha\left( {1 + n^{3}} \right)} + {\beta n_{+}}} \right\rbrack b} + {\left\lbrack {{\gamma\left( {1 + n^{3}} \right)} + {\delta n_{+}}} \right\rbrack c^{*}}}{a\sqrt{b\left( {1 + n^{3}} \right)}} = {{\left( {1 - {\frac{\left\lbrack {{4n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}\left( {1 + n^{3}} \right)}} \right.}{2\left( {1 + n^{3}} \right)}d\xi}} \right)\left( {1 - {\frac{\left\lbrack {{2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}}} \right\rbrack}{2\left( {1 + n^{3}} \right)}d\xi}} \right)\left( {1 + \left\lbrack {\frac{\left( {{2\overset{\sim}{\alpha}{n^{3}\left( {1 + n^{3}} \right)}} + {2\overset{\sim}{\beta}n_{+}} + {2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {{{Re}\left( {{\overset{\sim}{\gamma}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {2{in}_{+}n_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}}} \right.}{2\left( {1 + n^{3}} \right)}d\xi} \right\rbrack} \right)} = {{1 + {{\frac{1}{2\left( {1 + n^{3}} \right)}\left\lbrack {{{- 4}n^{3}{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} - {2{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}\left( {1 + n^{3}} \right)} - {2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)} - {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}} + {2\overset{\sim}{\alpha}{n^{3}\left( {1 + n^{3}} \right)}} + {2\overset{\sim}{\beta}n_{+}} + {2{{Re}\left( \overset{\sim}{\alpha} \right)}\left( {1 + n^{3}} \right)^{2}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {{{Re}\left( {{\overset{\sim}{\gamma}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {2{in}_{+}n_{-}{{Im}\left( \overset{\sim}{\alpha} \right)}}} \right\rbrack}d\xi}} = {1 + {{\frac{1}{2\left( {1 + n^{3}} \right)}\left\lbrack {{{- 2}{{Re}\left( {{\overset{\sim}{\beta}n_{+}} + {\overset{\sim}{\gamma}n_{-}}} \right)}\left( {1 + n^{3}} \right)} - {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}} + {2\overset{\sim}{\beta}n_{+}} + {2{{Re}\left( {\overset{\sim}{\beta}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {{{Re}\left( {{\overset{\sim}{\gamma}}^{*}n_{+}} \right)}\left( {1 + n^{3}} \right)} + {2{i\left( {1 + n^{3}} \right)}{{Im}\left( \overset{\sim}{\alpha} \right)}}} \right\rbrack}d\xi}}}}}} & ({S48})\end{matrix}$

Therefore, we can conclude Eq. (S47) has the form

$\begin{matrix}{{{{\frac{1}{a\sqrt{b\left( {1 + n^{3}} \right)}}\left\lbrack {{\alpha\left( {1 + n^{3}} \right)} + {\beta n}_{+}} \right\rbrack}b} + {\left\lbrack {{\gamma\left( {1 + n^{3}} \right)} + {dn}_{+}} \right\rbrack c^{*}}} = {1 + {i\left\lbrack {{{{Re}\left( \overset{\sim}{\beta} \right)}\frac{n^{\hat{2}}}{1 + n^{\hat{3}}}} + {{{Im}\left( \overset{\sim}{\beta} \right)}\frac{n^{\hat{1}}}{1 + n^{\hat{3}}}} + {{{Im}\left( \overset{\sim}{\alpha} \right)}d{\xi.}}} \right.}}} & ({S49})\end{matrix}$

By the definition, the infinitesimal Wigner angle is

$\begin{matrix}{\overset{\sim}{\psi} = {{2{{Im}\left( \overset{\sim}{\alpha} \right)}} + {\frac{2n^{\hat{1}}}{1 + n^{\hat{3}}}{{Im}\left( \overset{\sim}{\beta} \right)}} + {\frac{2n^{\hat{2}}}{1 + n^{\hat{3}}}{{Im}\left( \overset{\sim}{\gamma} \right)}}}} & ({S50})\end{matrix}$

Tetrads for Stationary Observer

The tetrads, e_(a) ^(μ)(x), are defined as (5, 6)g _(μv)(x)e _(a) ^(μ)(x)·e _(b) ^(v)(x)=η_(ab).  (S51)

For a stationary observer (Bob), his local frame is definedmathematically with the following tetrads in Schwarzschild spacetime,(e ₀)^(μ)(x)=(e _(t))^(μ)(x)=(1/(1−r _(s) /r)^(1/2),0,0,0)(e ₁)^(μ)(x)=(e _(r))^(μ)(x)=(0,(1−r _(s) /r)^(1/2),0,0)(e ₂)^(μ)(x)=(e _(θ))^(μ)(x)=(0,0,1/r,0)(e ₃)^(μ)(x)=(e _(ϕ))^(μ)(x)=(0,0,0,1/r).  (S52)

This tetrad represents a static local inertial frame at each point,since all the components are independent of time and spatial componentsof the timelike tetrad, e_(i) ^(t) (x) where i=1, 2, 3, and the timecomponents of spacelike tetrads, e_(o) ^(a) (x) where α=r, θ, φ, arezero. The corresponding ILLT(Infinitesimal Local Lorentz Transformation)matrix is given by

$\begin{matrix}{\left( \omega_{b}^{a} \right) = {\left( \begin{matrix}0 & {- \frac{k^{t}r_{s}}{2r^{2}}} & 0 & 0 \\{- \frac{k^{t}r_{s}}{2r^{2}}} & 0 & {- {k^{\theta}\left( {1 - \frac{r_{s}}{r}} \right)}^{1/2}} & 0 \\0 & {k^{\theta}\left( {1 - \frac{r_{s}}{r}} \right)}^{1/2} & 0 & 0 \\0 & 0 & 0 & 0\end{matrix}\text{⁠} \right).}} & ({S53})\end{matrix}$

It is easy to see all the parameters, defined in Eq. (S34), are realwith this ILLT matrix.

Therefore, Bob at rest cannot observe non-zero Wigner rotation angle.

Tetrads for Free Falling Observer with Zero Angular Momentum

The observer starts from rest at infinity and fall radially inward. Theobserver's energy and angular momentum, defined in equation (S5), are 1and 0, respectively. [Hartle] Thus, the timelike component of 4-velocityvector is (1−r_(s)/r)⁻¹ and the component of ϕ-direction of 4-veclocityvector is zero. The component of θ-direction is also zero, since weassume that observer travels in the plane θ=π/2. By substituting thecomponent of t-direction into the following equation,

$\begin{matrix}{{{{{- \left( {1 - \frac{r_{s}}{r}} \right)}\left( \frac{dt}{d\xi} \right)^{2}} + {\left( {1 - \frac{r_{s}}{r}} \right)^{- 1}\left( \frac{dr}{d\xi} \right)^{2}}} = {- 1}},} & ({S54})\end{matrix}$we can obtain the component of r-direction. In this case, the ILLTmatrix is given by

$\begin{matrix}{\left( {S55} \right){\left( \omega_{b}^{a} \right) = {\text{⁠}{{\left( \begin{matrix}0 & {{- \frac{k^{t}r_{s}}{2r^{2}}} - {\sqrt{\frac{r_{s}}{r}}\frac{k^{r}}{2{r\left( {1 - \frac{r_{s}}{r}} \right)}}}} & {k^{\theta}\sqrt{\frac{r_{s}}{r}}} & 0 \\{{- \frac{k^{t}r_{s}}{2r^{2}}} - {\sqrt{\frac{r_{s}}{r}}\frac{k^{r}}{2{r\left( {1 - \frac{r_{s}}{r}} \right)}}}} & 0 & k^{\theta} & 0 \\{k^{\theta}\sqrt{\frac{r_{s}}{r}}} & {- k^{\theta}} & 0 & 0 \\0 & 0 & 0 & 0\end{matrix}\text{⁠} \right).}}}}} & \end{matrix}$

Tetrads for Free Falling Observers with Non-Zero Angular Momentum I(Spiral Orbit)

In this case, we also assume that e=1[Hartle]. The observer, who isfalling free with angular momentum has the 4-velocity vector such that

$\begin{matrix}{{u(\tau)} = {{\left( e_{\hat{t}} \right)^{\mu}(x)} = \left( {{1/\left( {1 - {r_{s}/r}} \right)},u^{r},0,\frac{l}{r^{2}}} \right)}} & ({S56})\end{matrix}$ $\begin{matrix}{where} & \end{matrix}$ $\begin{matrix}{u^{r} = {- {\left( {\frac{r_{s}}{r} - {\frac{l^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}} \right)^{1/2}.}}} & ({S57})\end{matrix}$

One of the sets of the tetrads for free falling observer with non-zeroangular momentum is as follows

$\begin{matrix}{{{\left( e_{\hat{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {\frac{1}{\left( {1 - \frac{r_{s}}{r}} \right)},{{- \sqrt{\frac{r_{s}}{r}}}\cos{\Theta(r)}},{{- \frac{1}{r}}\sqrt{\frac{r_{s}}{r}}\frac{\sin{\Theta(r)}}{1 - \frac{r_{s}}{r}}},0} \right)}}\text{}{{\left( e_{\hat{1}} \right)^{\mu}(x)} = \left( {{{- \sqrt{\frac{r_{s}}{r}}}\frac{1}{\left( {1 - \frac{r_{s}}{r}} \right)}},{\cos{\Theta(r)}},{\frac{1}{r}\frac{\sin{\Theta(r)}}{\sqrt{1 - \frac{r_{s}}{r}}}},0} \right)}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = \left( {0,{{- \sin}{\Theta(r)}\sqrt{1 - \frac{r_{s}}{r}}},{\frac{1}{r}\cos{\Theta(r)}},0} \right)}{{\left( e_{\hat{3}} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{1/r}} \right)}}{{{{where}\cos{\Theta(r)}} = \sqrt{\left( {1 - {\frac{l^{2}}{{rr}_{s}}\left( {1 - \frac{r_{s}}{r}} \right)}} \right)}},{{\sin{\Theta(r)}} = {- {\sqrt{\frac{l^{2}}{{rr}_{s}}\left( {1 - \frac{r_{s}}{r}} \right)}.}}}}} & ({S58})\end{matrix}$With the orthogonality condition, these tetrads can be rewritten moregenerally as

$\begin{matrix}{{{\left( e_{\hat{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {\frac{1}{\left( {1 - \frac{r_{s}}{r}} \right)},{{- \sqrt{\frac{r_{s}}{r}}}\cos{\Theta(r)}},{{- \frac{1}{r}}\sqrt{\frac{r_{s}}{r}}\frac{\sin{\Theta(r)}}{1 - \frac{r_{s}}{r}}},0} \right)}}{{\left( e_{\hat{1}} \right)^{\mu}(x)} = \left( {{{- \sqrt{\frac{r_{s}}{r}}}\frac{\cos{\overset{\sim}{\Theta}(r)}}{\left( {1 - \frac{r_{s}}{r}} \right)}},{{\cos{\Theta(r)}\cos{\overset{\sim}{\Theta}(r)}} - {\sin{\Theta(r)}\sin{\overset{\sim}{\Theta}(r)}\sqrt{1 - \frac{r_{s}}{r}}}},{{\frac{1}{r}\frac{\sin{\Theta(r)}\cos{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{r_{s}}{r}}}} + {\frac{1}{r}\cos{\Theta(r)}\sin{\overset{\sim}{\Theta}(r)}}},0} \right)}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = \left( {{\sqrt{\frac{r_{s}}{r}}\frac{\sin{\overset{\sim}{\Theta}(r)}}{\left( {1 - \frac{r_{s}}{r}} \right)}},{{\cos{\Theta(r)}\frac{\sin\overset{\sim}{\Theta}(r)}{\left( {1 - \frac{r_{s}}{r}} \right)}} - {\sin{\Theta(r)}\cos{\overset{\sim}{\Theta}(r)}\sqrt{1 - \frac{r_{s}}{r}}}},{{\frac{1}{r}\cos{\Theta(r)}\cos{\overset{\sim}{\Theta}(r)}} - {\frac{1}{r}\frac{\sin{\Theta(r)}\sin{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{r_{s}}{r}}}}},0} \right)}{{\left( e_{\hat{3}} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{1/r}} \right)}}} & ({S59})\end{matrix}$

To get non-spinning frame, tetrads should be parallel transported. Thus,the following condition must holdu ^(μ)∇_(μ) e _(â) ^(t)=0.  (S60)

In other words,

$\begin{matrix}{{\frac{d}{dr}{\overset{\sim}{\Theta}(r)}} = {- {\frac{l_{obs}}{2r^{2}u^{r}}.}}} & ({S61})\end{matrix}$

Since sin Θ(r) is approximately the same as

${- \sqrt{\frac{l^{2}}{{rr}_{s}}}},$we can deduce the following relation:

$\begin{matrix}{{\frac{d}{dr}{\Theta(r)}▯\frac{l_{obs}}{2r^{2}u^{r}}} = {\frac{d}{dr}{{\overset{\sim}{\Theta}(r)}.}}} & ({S62})\end{matrix}$

In other words, these tetrads can be written approximately as

$\begin{matrix}{{{\left( e_{\hat{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {\frac{1}{\left( {1 - \frac{r_{s}}{r}} \right)},{{- \sqrt{\frac{r_{s}}{r}}}\cos{\Theta(r)}},{{- \frac{1}{r}}\sqrt{\frac{r_{s}}{r}}\frac{\sin{\Theta(r)}}{\sqrt{1 - \frac{r_{s}}{r}}}},0} \right)}}{{\left( e_{\hat{1}} \right)^{\mu}(x)} = {{\left( e_{r} \right)^{\mu}(x)}==\left( {{{- \sqrt{\frac{r_{s}}{r}}}\frac{\cos{\Theta(r)}}{\left( {1 - \frac{r_{s}}{r}} \right)}},{{\cos 2{\Theta(r)}} + {\sin^{2}{\Theta(r)}\left( {1 - \sqrt{\left( {1 - \frac{r_{s}}{r}} \right)}} \right)}},{\frac{1}{r}\frac{\sin 2{\Theta(r)}}{2}\left( {1 + \frac{1}{\sqrt{1 - \frac{r_{s}}{r}}}} \right)},0} \right)}}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = {{\left( e_{\theta} \right)^{\mu}(x)}==\left( {{\sqrt{\frac{r_{s}}{r}}\frac{\sin{\Theta(r)}}{\left( {l - \frac{r_{s}}{r}} \right)}},{{- \frac{\sin 2{\Theta(r)}}{2}}\left( {1 + \sqrt{\left( {1 - \frac{r_{s}}{r}} \right)}} \right)},{{\frac{1}{r}\cos 2{\Theta(r)}} + {\frac{1}{r}\sin^{2}{\Theta(r)}\left( {1 - \frac{1}{\sqrt{1 - \frac{r_{s}}{r}}}} \right)}},0} \right)}}{{\left( e_{\hat{3}} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{1/r}} \right)}}} & ({S63})\end{matrix}$

By ignoring the first order of

$\frac{r_{s}}{r},$the physical meaning becomes much clearer. The tetrads can be written as

$\begin{matrix}{{{\left( e_{\overset{\hat{}}{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {1,\ {{- \sqrt{\frac{r_{s}}{r}}}\cos{\Theta(r)}},{{- \frac{1}{r}}\sqrt{\frac{r_{s}}{r}}\sin{\Theta(r)}},0} \right)}}{{\left( e_{\hat{1}} \right)^{\mu}(x)} = {{\left( e_{r} \right)^{\mu}(x)}==\left( {{{- \sqrt{\frac{r_{s}}{r}}}\cos{\Theta(r)}},{\cos 2{\Theta(r)}},{\frac{1}{r}\sin 2{\Theta(r)}},0} \right)}}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = {{{\left( e_{\theta} \right)^{\mu}(x)}=={{\left( {{\sqrt{\frac{r_{s}}{r}}\sin{\Theta(r)}},{{- \sin}2{\Theta(r)}},{\frac{1}{r}\cos 2{\Theta(r)}},0} \right).\left( e_{\hat{3}} \right)^{\mu}}(x)}} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{1/r}} \right)}}}} & ({S64})\end{matrix}$

Thus, we can conclude that if the tetrads, which are non-spinning andfree falling with non-zero angular momentum, are projected into3-dimension space, they rotate by 2 times of Θ(r) when observer's movingdirection rotates only by Θ(r). By the first order of angular momentum,l, and

$\frac{r_{s}}{r}$approximation, tetrads become

$\begin{matrix}{{{\left( e_{0} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {\frac{1}{\left( {1 - {r_{s}/r}} \right)},{- \sqrt{\frac{r_{s}}{r}}},\frac{l}{r^{2}},0} \right)}}{{\left( e_{1} \right)^{\mu}(x)} = {{\left( e_{r} \right)^{\mu}(x)} = \left( {{{- \sqrt{\frac{r_{s}}{r}}}\frac{1}{1 - {r_{s}/r}}},1,\frac{{- 2}l}{\sqrt{r_{s}r^{3}}},0} \right)}}{{\left( e_{2} \right)^{\mu}(x)} = {{\left( e_{\theta} \right)^{\mu}(x)} = \left( {\frac{- l}{r\left( {1 - {r_{s}/r}} \right)},\frac{l\left( {2 - \frac{r_{s}}{r}} \right)}{\sqrt{r_{s}r}},\frac{1}{r},0} \right)}}{{\left( e_{3} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{1/r}} \right)}}} & ({S65})\end{matrix}$

The ILLT matrix is then given by

$\begin{matrix}{\left( \lambda_{d}^{c} \right) = \begin{pmatrix}0 & {{{- \varepsilon_{r}^{2}}\frac{k^{t}r_{s}}{2r^{2}}} - {\varepsilon_{r}\sqrt{\frac{r_{s}}{r}}\frac{k^{r}}{2r}}} & 0 & {\varepsilon_{r}k^{\phi}\sqrt{\frac{r_{s}}{r}}} \\{{{- \varepsilon_{r}^{2}}\frac{k^{t}r_{s}}{2r^{2}}} - {\varepsilon_{r}\sqrt{\frac{r_{s}}{r}}\frac{k^{r}}{2r}}} & 0 & {\varepsilon_{l}^{2}\frac{k^{r}l}{\sqrt{r^{3}r_{s}}}} & k^{\phi} \\0 & {{- \varepsilon_{l}^{2}}\frac{k^{r}l}{\sqrt{r^{3}r_{s}}}} & 0 & {\varepsilon_{l}^{2}\frac{2k^{\phi}l}{\sqrt{{rr}_{s}}}} \\{\varepsilon_{r}k^{\phi}\sqrt{\frac{r_{s}}{r}}} & {- k^{\phi}} & {{- \varepsilon_{l}^{2}}\frac{2k^{\phi}l}{\sqrt{{rr}_{s}}}} & 0\end{pmatrix}} & ({S66})\end{matrix}$

Thus, the observer, falling free with non-zero angular momentum, seesthe non-zero Wigner angle as

$\begin{matrix}{\overset{\sim}{\psi} = {{- \varepsilon_{l}^{2}}\frac{k^{r}l}{\sqrt{r^{3}r_{s}}}}} & ({S67})\end{matrix}$

Tetrads for Free Falling Observers with Non-Zero Angular Momentum II(Circular Orbit)

In this case, we also assume that e=1 for the simplicity of thecalculations. The observer who is falling freely with angular momentumwith e=1 has the 4-velocity vector such that

$\begin{matrix}{{u(\tau)} = {{\left( e_{\hat{i}} \right)^{\mu}(x)} = \left( {{1/\left( {1 - {r_{s}/r}} \right)},u^{r}\ ,0,\frac{l}{r^{2}}} \right)}} & ({S68})\end{matrix}$ $\begin{matrix}{where} & \end{matrix}$ $\begin{matrix}{u^{r} = {- {\left( {\frac{r_{s}}{r} - {\frac{l^{2}}{r^{2}}\left( {1 - \frac{r_{s}}{r}} \right)}} \right)^{1/2}.}}} & ({S69})\end{matrix}$

In addition, by forcing the radial component of 4-velocity vector to benull, we can get

$\begin{matrix}{{{\left( e_{\overset{\hat{}}{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {\frac{1}{\sqrt{1 - \frac{3r_{s}}{2r}}},{0\frac{1}{r}\sqrt{\frac{r_{s}}{2r}}\frac{1}{\sqrt{1 - \frac{3r_{s}}{2r}}}},0} \right)}}{{\left( e_{\overset{\hat{}}{1}} \right)^{\mu}(x)} = {\left( e_{r} \right)^{\mu}(x)\left( {{{- \sqrt{\frac{r_{s}}{2r}}}\frac{\sin{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{3r_{s}}{2r}}\sqrt{1 - \frac{r_{s}}{r}}}},{\sqrt{1 - \frac{r_{s}}{r}}\cos{\overset{\sim}{\Theta}(r)}},{{- \frac{1}{r}}\frac{\sqrt{1 - \frac{r_{s}}{r}}\sin{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{3r_{s}}{2r}}}},0} \right)}}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = {\left( e_{\theta} \right)^{\mu}(x)\left( {{\sqrt{\frac{r_{s}}{2r}}\frac{\cos{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{3r_{s}}{2r}}\sqrt{1 - \frac{r_{s}}{r}}}},{\sqrt{1 - \frac{r_{s}}{r}}\sin{\overset{\sim}{\Theta}(r)}},\sqrt{1 - \frac{r_{s}}{r}},{\frac{1}{r}\frac{\sqrt{1 - \frac{r_{s}}{r}}\cos{\overset{\sim}{\Theta}(r)}}{\sqrt{1 - \frac{3r_{s}}{2r}}}},0} \right)}}{{\left( e_{\hat{3}} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{\csc{\theta/r}}} \right)}}} & ({S70})\end{matrix}$

Likewise, by adding non-spinning condition, we can get

$\begin{matrix}{{{\overset{˜}{\Theta}(r)} = {\sqrt{1 - \frac{3r_{s}}{2r}}\left( {\theta - \theta_{0}} \right)}}.} & ({S71})\end{matrix}$

By ignoring the first order of rs/r, tetrads are written as

$\begin{matrix}{{{\left( e_{\overset{\hat{}}{0}} \right)^{\mu}(x)} = {{\left( e_{t} \right)^{\mu}(x)} = \left( {1,{0\frac{1}{r}\sqrt{\frac{r_{s}}{2r}}},0} \right)}}{{\left( e_{\overset{\hat{}}{1}} \right)^{\mu}(x)} = {\left( e_{r} \right)^{\mu}(x)\left( {{{- \sqrt{\frac{r_{s}}{2r}}}\sin{\overset{\sim}{\Theta}(r)}},{\cos{\overset{\sim}{\Theta}(r)}},{{- \frac{1}{r}}\sin{\overset{\sim}{\Theta}(r)}},0} \right)}}{{\left( e_{\hat{2}} \right)^{\mu}(x)} = {\left( e_{\theta} \right)^{\mu}(x)\left( {{\sqrt{\frac{r_{s}}{2r}}\cos{\overset{\sim}{\Theta}(r)}},{\sin{\overset{\sim}{\Theta}(r)}},{{\frac{1}{r}\cos{\overset{\sim}{\Theta}\left( {r,0} \right)}\left( e_{\hat{3}} \right)^{\mu}(x)} = {{\left( e_{\phi} \right)^{\mu}(x)} = \left( {0,0,0,{\csc{\theta/r}}} \right)}}} \right.}}} & ({S72})\end{matrix}$where {tilde over (Θ)}(r)=(θ−θ₀).

That is, by projecting tetrads into 3-dimensional space, one can seethat tetrads rotate by −θ when observer moves by rθ. This is quite anon-relativistic effect, since the rotation, induced by paralleltransportation condition, is just compensation of the effect ofspherical coordinate system. In FIG. S1, the components, classical andquantum, of the Wigner rotation for an free-falling observer withnon-zero angular momentum for the circular orbit. We set angularmomentum of observers as

$0.4{\sqrt{r_{s}r_{earth}}.}$

From Wigner Angle to the Rotation Angle of Polarization

$\begin{matrix}{{{\varepsilon_{\pm}^{\prime\mu}\left( \overset{\hat{}}{k^{\prime}} \right)} \equiv {{D(\Lambda)}{\varepsilon_{\pm}^{\mu}\left( \overset{\hat{}}{k^{\prime}} \right)}}} = {{R\left( {\Lambda\overset{\hat{}}{k}} \right)}{R_{z}\left( {\psi\left( {\Lambda,\overset{\rightarrow}{n}} \right)} \right)}{R\left( \overset{\hat{}}{k} \right)}^{- 1}{\varepsilon_{\pm}^{\mu}\left( \overset{\hat{}}{k} \right)}}} & ({S73})\end{matrix}$ $\begin{matrix}{{\varepsilon_{\phi}^{\mu}\left( \overset{\hat{}}{k} \right)} = {{\frac{1}{\sqrt{2}}\left( {{e^{i\phi}{\varepsilon_{+}^{\mu}\left( \overset{\hat{}}{k} \right)}} + {e^{{- i}\phi}{\varepsilon_{-}^{\mu}\left( \overset{\hat{}}{k} \right)}}} \right)} = {{{R\left( \overset{\hat{}}{k} \right)}\ \begin{bmatrix}0 \\{\cos\phi} \\{\sin\phi} \\0\end{bmatrix}} \equiv {{R\left( \overset{\hat{}}{k} \right)}{{\overset{˜}{\varepsilon}}_{\phi}^{\mu}\left( \overset{\hat{}}{z} \right)}}}}} & ({S74})\end{matrix}$ $\begin{matrix}{{\varepsilon_{\phi}^{\prime\mu}\left( {\overset{\hat{}}{k}\prime} \right)} = {{\frac{1}{\sqrt{2}}\left( {{e^{i\phi^{\prime}}{\varepsilon_{+}^{\mu}\left( \overset{\hat{}}{k^{\prime}} \right)}} + {e^{{- i}\phi^{\prime}}{\varepsilon_{-}^{\mu}\left( \overset{\hat{}}{k^{\prime}} \right)}}} \right)} = {{{R\left( {\Lambda\overset{\hat{}}{k}} \right)}\ \begin{bmatrix}0 \\{\cos\phi^{\prime}} \\{\sin\phi^{\prime}} \\0\end{bmatrix}} \equiv {{R\left( {\Lambda\overset{\hat{}}{k}} \right)}{{\overset{\sim}{\varepsilon}}_{\phi^{\prime}}^{\prime\mu}\left( \overset{\hat{}}{z} \right)}}}}} & ({S75})\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{\varepsilon}}_{\phi^{\prime}}^{\prime\mu}\left( \hat{z} \right)} = {{R_{z}\left( {\psi\left( {\Lambda,\overset{\rightarrow}{n}} \right)} \right)}{{\overset{\sim}{\varepsilon}}_{\phi}^{\mu}\left( \hat{z} \right)}}} & ({S76})\end{matrix}$ $\begin{matrix}{{\therefore\phi^{\prime}} = {\phi + \psi}} & ({S77})\end{matrix}$

This result tells us Wigner angle is the same with polarization rotationangle only in the standard frame. However, our 3-axis is not parallel tothe wave vector. Therefore, by decomposing Wigner rotation into otherrotations, we can get the rotation angle about wave vector, induced byWigner rotation, which corresponds to the angle of polarizationrotation. If a rotation is denoted by (n,φ), the rotation can bedecomposed into three consecutive rotations denoted by (n_(i), φ_(i))with the axis unit vectors, n_(i), and the corresponding rotationangles, φ_(i). When mutually orthogonal axes are considered, thefollowing relation holds[Decomposition of a Finite Rotation . . . ],

$\begin{matrix}{{\sin\varphi_{3}} = \frac{{b_{1}{b_{2}\left( {1 - {\cos\varphi}} \right)}} + {b_{3}\sin\varphi}}{\cos\varphi_{2}}} & ({S78})\end{matrix}$where b_(i)=n·n_(i). Since we deal with infinitesimal angles, we can getφ₃ =b ₃φ  (S79)

In our case, the unit vectors, n,n_(i), are defined by

$\begin{matrix}{{n_{\hat{i}} = \frac{\partial}{\partial x^{\hat{i}}}},{n = {\frac{k^{\hat{i}}}{k^{\hat{t}}}\frac{\partial}{\partial x^{\hat{i}}}}},} & ({S80})\end{matrix}$

The coefficient, b_(i), is obtained from the definition

$\begin{matrix}{b_{i} = {{n \cdot n_{i}} = {\frac{k^{\hat{i}}}{k^{\hat{t}}} = {\frac{\eta^{\hat{i}\hat{i}}{e_{\hat{i}}^{\mu}\left( {g_{\mu\mu}k^{\mu}} \right)}}{\eta^{\overset{\hat{}}{0}\overset{\hat{}}{0}}{e_{\hat{i}}^{\mu}\left( {g_{\mu\mu}k^{\mu}} \right)}} \cong {\frac{{rk}^{\phi}}{k^{t}}\left( {{i = 1},2,3} \right)}}}}} & ({S81})\end{matrix}$

Therefore, Eq. (S78) becomes

$\begin{matrix}{{{polarization}{angle}} = {\frac{{rk}^{\phi}}{k^{t}} \times \left( {{Wigner}{rotation}{angle}} \right)}} & ({S82})\end{matrix}$

The corresponding infinitesimal polarization rotation angle is

$\begin{matrix}{\overset{\sim}{\psi} = {{- \varepsilon_{l}^{2}}{\frac{k^{\phi}k^{r}l}{k^{t}\sqrt{{rr}_{s}}}.}}} & ({S83})\end{matrix}$

Torsion-Free

Every tetrads used in this paper is torsion-free. It is easy to checkwhether the local frame described by tetrads has torsion or not bycalculating components of torsion tensor in the local basis, which aredefined as:T ^(â) _({circumflex over (b)}ĉ)=Γ_({circumflex over (b)}ĉ)^(â)−Γ_(ĉ{circumflex over (b)}) ^(â) −c _({circumflex over (b)}ĉ) ^(â),where Γ_({circumflex over (b)}ĉ) ^(â) =e ^(â) _(μ) e_({circumflex over (b)}) ^(v)∇_(v) e _(ĉ) ^(μ and c)_(â{circumflex over (b)}) ^(ĉ)=Γ^(ĉ) _(â{circumflex over (b)})−Γ^(ĉ)_({circumflex over (b)}â).  (S84)

Method for Measuring a Distance to a Satellite

FIG. 6 is a flow chart showing a method for measuring a distance to asatellite according to an exemplary embodiment of the present invention.

Referring to FIG. 6 , according to a method of measuring a distance to asatellite, according to an exemplary embodiment of the presentinvention, an electronic device receive a linearly polarized photon fromand angular momentum per unit mass of the satellite the satellite (stepS110). For example, the electronic device may be a user terminal such asa navigation system for a vehicle, a smartphone, etc.

Then, the electronic device measures an amount of rotation of thepolarized photon, the rotation being induced by a space-time warpage dueto gravity (step S120).

Then, the electronic device calculates a distance to the satellite byusing the rotation amount of the polarized photon and the angularmomentum per unit mass of the satellite (step S130).

For example, the distance to the satellite may be calculated by thefollowing equation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

The above equation is explained above in detail, so that any furtherexplanation will be omitted.

Method for Measuring a Position

FIG. 7 is a flow chart showing a method for measuring a positionaccording to an exemplary embodiment of the present invention.

Referring to FIG. 7 , according to a method of measuring a locationaccording to an exemplary embodiment of the present invention, anelectronic device receives, from at least three or more satellites, apolarized photon of each satellite and angular momentum per unit mass ofthe satellites (step S210). Preferably, it is desirable to receivelinearly polarized photons of each satellite and angular momentum perunit mass of the satellite from four or more satellites in considerationof the height of the electronic device. That is, when it is assumed thatthe electronic device exists on a perfect spherical surface, threesatellites are sufficient, but since the Earth's surface has elevations,at least four or more satellites are required to accurately measure alocation.

Then, the electronic device measures an amount of rotation of thepolarized photon of each satellite, the rotation being induced bywarpage of space-time due to gravity (step S220), and calculates adistance to each satellite by using a rotation amount of polarization ofeach satellite and an angular momentum per unit mass of each satellite(step S230).

The distance to each satellite may be calculated by the followingequation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

Then, the electronic device calculates a position relative to each ofthe satellites by using the distance to each of the satellites (stepS240). This process will be explained in detail, referring to FIG. 9 .

FIG. 8 is a flow chart showing a method for measuring a locationaccording to another exemplary embodiment of the present invention. Inthe previous embodiment of FIG. 7 , only the position of the electronicdevice, which is relative to each satellite can be obtained, but theabsolute position (or coordination) of the electronic device can beobtained in the present embodiment by using the absolute coordination.

Referring to FIG. 7 and FIG. 8 , the electronic device further receivesa coordinated of each of the satellites from each of the satellites bythe electronic device (step S310), when receiving the polarized photonand angular momentum per unit mass of the satellites (step S210).

Further, the electronic device calculates a position of the electronicdevice (step S340), when calculating a position relative to each of thesatellites (step S240).

Hereinafter, a principle of measuring the relative position and theabsolute position of the electronic device will be described withreference to FIG. 9 below.

FIG. 9 is a diagram showing a method of measuring relative positionsbetween satellites using distances from three satellites.

Referring to FIG. 9 , when a user terminal is separated by R1 from thefirst satellite (Sat1), by R2 from the second satellite (Sat2), andseparated by R3 from the third satellite (Sat3), the location of theuser terminal is fixed to the point P.

In FIG. 9 , it will be understood that the location of the user terminalis fixed in the case of three satellites in a two-dimensional space withno height, but four satellites are required if there is athree-dimensional space with a height.

User Terminal

FIG. 10 is a block diagram showing a user terminal for implementing amethod for measuring a location according to the present invention.

Referring to FIG. 10 , a user terminal 100 according to an exemplaryembodiment of the present invention comprises a photon reception unit110, a satellite information reception unit 120, a polarization rotationmeasurement unit 130, and a calculation unit 140.

The photon reception unit 110 receives a polarized photon of eachsatellite from at least three or more satellites.

The satellite information reception unit 120 receives an angularmomentum per unit mass of each satellite from the at least three or moresatellites.

The polarization rotation measurement unit 130 measures a rotationamount of the polarized photon received by the photon receiving unit 110from each satellite.

The calculation unit 140 calculates a distance to each of the satellitesby using the rotation amount of the polarized photon, and the angularmomentum per unit mass of each of the satellites, and calculatesrelative position of the user terminal by using the distance to each ofthe satellites as shown in FIG. 7 . For example, the calculation units140 may calculate the distance to each of the satellites by thefollowing equation,

${{\sin{\Theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}},$

wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ is the Schwarzschild radius of the Earth.

Preferably, the satellite information reception unit 120 may furtherreceive a coordinate of each of the satellites from the each of thesatellites, and the calculation unit 140 may calculate a coordinates ofthe user terminal by using the coordinate of each of the satellites.

As described above, according to the present invention, it is possibleto more accurately measure the distance between the satellite and theuser terminal, thereby improving accuracy in a location measuring systemsuch as GPS.

It will be apparent to those skilled in the art that variousmodifications and variation may be made in the present invention withoutdeparting from the spirit or scope of the invention. Thus, it isintended that the present invention cover the modifications andvariations of this invention provided they come within the scope of theappended claims and their equivalents.

What is claimed is:
 1. A method of measuring a distance to a satellite,which is performed by an electronic device, the method comprising:receiving a linearly polarized photon and angular momentum per unit massof the satellite from the satellite; measuring an amount of rotation ofthe polarized photon, the rotation being induced by a space-time warpagedue to gravity; and calculating a distance to the satellite by using therotation amount of the polarized photon and the angular momentum perunit mass of the satellite.
 2. The method of claim 1, wherein thedistance to the satellite is calculated by the following equation,${\sin{\theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}$wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ the Schwarzschild radius of the Earth. 3.A method of measuring a location comprising: receiving, by an electronicdevice, from at least three or more satellites, a polarized photon ofeach satellite and angular momentum per unit mass of the satellites;measuring, by the electronic device, an amount of rotation of thepolarized photon of each satellite, the rotation being induced bywarpage of space-time due to gravity; calculating, by the electronicdevice, a distance to each satellite by using a rotation amount ofpolarization of each satellite and an angular momentum per unit mass ofeach satellite; and calculating a position relative to each of thesatellites by the electronic device by using the distance to each of thesatellites.
 4. The method of claim 3, wherein the distance to eachsatellite is calculated by the following equation,${\sin{\theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}$wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ the Schwarzschild radius of the Earth. 5.The method of claim 3, wherein the electronic device further receives acoordinated of each of the satellites from each of the satellites whenreceiving the polarized photon and angular momentum per unit mass of thesatellites, and the electronic device further calculates a position ofthe electronic device when calculating a position relative to each ofthe satellites.
 6. A user terminal comprising: a photon reception unitreceiving a polarized photon of each satellite from at least three ormore satellites; a satellite information reception unit receiving anangular momentum per unit mass of each satellite from the at least threeor more satellites; a polarization rotation measurement unit measuring arotation amount of the polarized photon received by the photon receivingunit from each satellite; and a calculation unit calculating a distanceto each of the satellites by using the rotation amount of the polarizedphoton and the angular momentum per unit mass of each of the satellites,and calculating relative position of the user terminal by using thedistance to each of the satellites.
 7. The user terminal of claim 6,wherein the calculation units calculates the distance to each of thesatellites by the following equation,${\sin{\theta(r)}} \cong {{- \frac{l_{obs}}{\sqrt{{rr}_{s}}}}\sqrt{1 - \frac{r_{s}}{r}}}$wherein ‘2Θ’ is the rotation amount of polarized photon, ‘l_(obs)’ isthe angular momentum per unit mass of the satellite, ‘r’ is the distanceto the satellite, and ‘r_(s)’ the Schwarzschild radius of the Earth. 8.The user terminal of claim 6, wherein the satellite informationreception unit further receives a coordinate of each of the satellitesfrom the each of the satellites, and the calculation unit calculates acoordinates of the user terminal by using the coordinate of each of thesatellites.